Application of spectral filtering to discontinuous Galerkin methods on triangulations

Abstract

AbstractWe adapt the spectral viscosity (SV) formulation implemented as a modal filter to a discontinuous Galerkin (DG) method solving hyperbolic conservation laws on triangular grids. The connection between SV and spectral filtering, which is undertaken for the first time in the context of DG methods on unstructured grids, allows to specify conditions on the filter strength regarding time step choice and mesh refinement. A crucial advantage of this novel damping strategy is its low computational cost. We furthermore obtain new error bounds for filtered Dubiner expansions of smooth functions. While high order accuracy with respect to the polynomial degree N is proven for the filtering procedure in this case, an adaptive application is proposed to retain the high spatial approximation order. Although spectral filtering stabilizes the scheme, it leaves weaker oscillations. Therefore, as a postprocessing step, we apply the image processing technique of digital total variation (DTV) filtering in the new context of DG solutions and prove conservativity in the limit for this filtering procedure. Numerical experiments for scalar conservation laws confirm the designed order of accuracy of the DG scheme with adaptive modal filtering for polynomial degrees up to 8 and the viability of spectral and DTV filtering in case of shocks. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011